
Here \(F_\pi \)  is the pion decay constant and \(M_\pi \)  denotes the pion mass.
The LO \(^1 S_0\)  \(N\!N\)  contact coupling \(C\)  contains contributions from one-pion exchange as well as a contact interaction and scales as \(C \sim 4 \pi /(m_N Q)\) .
The split \(\hat{V} = \hat{V}_\pi + \hat{V}_S\)  implies that the \(\hat{T}\)  operator can be similarly separated into a pion-range and short-range contribution as follows [1]}
\(\hat{T} (E) &= \hat{T}_\pi (E) + \hat{T}_S (E)\,,\\\hat{T}_\pi (E) &= \hat{V}_\pi \left( I + \hat{G}_+^{(\pi )} (E) \, \hat{V}_\pi \right) = \hat{V}_\pi \ \left(I - \hat{G}^{(0)}_+ (E) \, \hat{V}_\pi \right)^{-1} \,,\\\hat{T}_S (E) &= \left( I + \hat{V}_\pi \hat{G}_+^{(\pi )} (E) \right) \left[ \hat{V}_S\, \left(I - \hat{G}^{(\pi )}_+ (E) \, \hat{V}_S \right)^{-1}\, \right]\left( I + \hat{G}_+^{(\pi )} (E) \, \hat{V}_\pi \right)\,,\) 
